On equivalence of unbounded metric spaces at infinity

Abstract

Let $(X,d)$ be an unbounded metric space. To investigate the asymptotic behavior of $(X,d)$ at infinity, one can consider a sequence of rescaling metric spaces $(X,\frac{1}{r_n}d)$ generated by given sequence $(r_n{)}_{n\in \mathbb{N}}$ of positive reals with $r_n\to \infty$. Metric spaces that are limit points of the sequence $(X,\frac{1}{r_n}d{)}_{n\in \mathbb{N}}$ will be called pretangent spaces to $(X,d)$ at infinity. We found the necessary and sufficient conditions under which two given unbounded subspaces of $(X,d)$ have the same pretangent spaces at infinity. In the case when $(X,d)$ is the real line with Euclidean metric, we also describe all unbounded subspaces of $(X,d)$ isometric to their pretangent spaces.

46 pages

References in corpus (1)

• Tangent spaces to metric spaces and to their subspaces